Which of the following is a countable set?

The correct answer is Set of all natural numbers.

Key Points

• A set is considered countable if it has the same cardinality as the set of natural numbers or if it is finite.
  • The set of all natural numbers (denoted as N = {1, 2, 3, ...}) is an example of a countable infinite set because each element can be uniquely paired with a natural number.
  • Countable sets can either be finite or have a one-to-one correspondence with the natural numbers.
• Other sets in the options are not countable:
  • The set of all irrational numbers is uncountable because it is a subset of the real numbers with infinite cardinality that cannot be paired with natural numbers.
  • The set of all values in the closed interval [0, 1] is uncountable because it contains all real numbers between 0 and 1, which form a continuum.
  • The set of all complex numbers is uncountable because it includes all possible combinations of real and imaginary numbers.

Additional Information

• In mathematics, the concept of countability is essential in set theory and helps classify the size of infinite sets.
• The set of integers and the set of rational numbers are also countable because they can be arranged in a sequence that corresponds to natural numbers.
• The concept of countability was formalized by mathematician Georg Cantor, who developed set theory.
• The study of uncountable sets, such as the real numbers, led to the development of advanced mathematical fields like topology and measure theory.